Fourier conjugates and momentum

[redtree]

Given that position and momentum are Fourier conjugates, what is the derivation for the equation $\hbar \vec{k} = m \vec{v}$, where momentum-space momentum is defined as $\hbar \vec{k}$ and position-space momentum is defined as $m \vec{v}$?

 

[PeterDonis]

Write down an eigenfunction in momentum space, i.e., a plane wave with a single value of $\vec{k}$. Then Fourier transform it into the position representation. What is the phase velocity of the resulting wave function in position space?

(Another way of seeing it: what is the representation of the momentum operator in position space?)

 

[redtree]

Just to make sure I understand: are you telling me to Fourier transform the following?:
$\langle \vec{k} |\phi \rangle = \phi(\vec{k})$

In general, I prefer to work in standard mathematical notation (as opposed to Dirac notation) so an explanation in that notation would be much better from my perspective. I hope that doesn't inconvenience you too much.

 

[redtree]

The momentum operator in position space is $-i \hbar \frac{\partial}{\partial \vec{x}}$ while in momentum space it is $\vec{k}$. How does that lead to $\hbar \vec{k} = m \vec{v}$?

 

[PeterDonis]

are you telling me to Fourier transform the following?

No, because you didn't follow your own advice:

I prefer to work in standard mathematical notation (as opposed to Dirac notation)

Which is a good idea for this problem. So write down the momentum eigenfunction in standard notation, then Fourier transform it to position space--or, equivalently, write down the momentum eigenfunction in the position representation. It will describe a plane wave. What is the phase velocity of that plane wave?

 

[redtree]

\begin{equation}
\begin{split}
\hat{\vec{p}} \psi(\vec{x})&= -i \hbar \frac{\partial}{\partial \vec{x}} \psi(\vec{x})
\end{split}
\end{equation}

Such that:
\begin{equation}
\begin{split}
\mathcal{F}\left[ -i \hbar \frac{ \partial}{\partial \vec{x}} \psi(\vec{x})\right]&=-i \hbar \frac{ \partial}{\partial \vec{x}} \int_{-\infty}^{\infty} d\vec{k} e^{2 \pi i \vec{k} \vec{x}} \phi(\vec{k})
\end{split}
\end{equation}

Not sure what I should do next?

 

[redtree]

By convention: the phase velocity ($\vec{v}_P$) is defined as follows, where $\vec{p}= \hbar \vec{k}$ and $E = \hbar \omega$
\begin{equation}
\begin{split}
\vec{v}_P &\doteq \frac{\omega}{\vec{k}}
\\
&=\frac{\omega}{\frac{\vec{p}}{\hbar}}
\\
&=\frac{\hbar \omega}{\vec{p}}
\\
&=\frac{E}{\vec{p}}
\end{split}
\end{equation}

 

[PeterDonis]

By convention: the phase velocity ($\vec{v}_P$) is defined as follows

Yes, the phase velocity is $E / p$. You know $p$ for the plane wave. What is $E$ for the plane wave? (Note: since you've defined momentum in position space as $m v$, you are implicitly assuming non-relativistic QM. So the energy of the plane wave should be the non-relativistic energy.)

 

[PeterDonis]

Not sure what I should do next?

In momentum space, what is $\phi(k)$? Bear in mind that we are talking about a plane wave, which only has nonzero amplitude for a single value of $k$.

 

[redtree]

Yes, the phase velocity is $E / p$. You know $p$ for the plane wave. What is $E$ for the plane wave? (Note: since you've defined momentum in position space as $mv$, you are implicitly assuming non-relativistic QM. So the energy of the plane wave should be the non-relativistic energy.)

I defined momentum in wavenumber (momentum) space as $\hbar \vec{k}$. I haven't actually defined momentum in position space at all. I am trying to understand how defining momentum in wavenumber space as I have leads to a measure of momentum space where $\vec{p} = m v$

Given:
\begin{equation}
\begin{split}
\hat{E}&= i \hbar \frac{\partial}{\partial t}
\end{split}
\end{equation}

Such that:
\begin{equation}
\begin{split}
\frac{\hat{E}}{\hat{\vec{p}}}&=\frac{\left(i \hbar \frac{\partial}{\partial t}\right)}{\left(-i \hbar \frac{\partial}{\partial \vec{x}} \right)}
\\
&=-\left( \frac{\partial }{\partial t} \right)\left( \frac{\partial \vec{x}}{\partial} \right)
\end{split}
\end{equation}

I don't think the partials $\partial$ cancel.

 

[PeterDonis]

Given:

^E​

$$\begin{equation} \begin{split} \hat{E}&= i \hbar \frac{\partial}{\partial t} \end{split} \end{equation}$$​

Where are you getting that from? Bear in mind that we are considering a momentum eigenstate; such a state is time-independent, so $\partial / \partial t$ of anything is zero. But it has a well-defined, nonzero energy. What is that energy? (Hint: look at the time-independent, non-relativistic Hamiltonian for a free particle.)

 

[PeterDonis]

I haven't actually defined momentum in position space at all.

Sure you have. Defining it in momentum space automatically defines it in position space, via the Fourier transform. The Fourier transform of a momentum eigenstate to position space is found in most QM textbooks. Or you could even try solving for the momentum space eigenstate in position space directly: just look for some function $f(x)$ that satisfies the eigenvalue equation $\hat{p} f(x) = - i \hbar \partial f / \partial x = \hbar \vec{k} f(x)$.

 

[redtree]

Sure you have. Defining it in momentum space automatically defines it in position space, via the Fourier transform. The Fourier transform of a momentum eigenstate to position space is found in most QM textbooks. Or you could even try solving for the momentum space eigenstate in position space directly: just look for some function $f(x)$ that satisfies the eigenvalue equation $\hat{p} f(x) = - i \hbar \partial f / \partial x = \hbar \vec{k} f(x)$.

That is true. Defining momentum in momentum space does imply a definition in position space via the Fourier transform. I misstated the problem. It's the equivalence between $\hbar \vec{k}$ and $m \vec{v}$ that I don't see.

 

[PeterDonis]

It's the equivalence between $\hbar \vec{k}$ and #m \vec{v}## that I don't see.

See my question in post #11. Also, I misstated my earlier question about phase velocity, I should have asked about group velocity, which is $dE / dp$, not $E / p$. But either way you want a formula for $E$ in terms of $p$ in order to answer the question.

 

[redtree]

In momentum space, what is $\phi(k)$? Bear in mind that we are talking about a plane wave, which only has nonzero amplitude for a single value of $k$.

My understanding of a plane wave is that it has non-zero amplitudes for many values of $\vec{k}$. Given:
\begin{equation}
e^{i x}=\cos(x) + i \sin(x)
\end{equation}If a wavefunction is zero for all values of $\vec{k}$, except a single value $\vec{k}_0$, then the I would assume the following is true:
\begin{equation}
\begin{split}
\phi(\vec{k})&= \delta(\vec{k}-\vec{k}_0)
\end{split}
\end{equation}

Where:
\begin{equation}
\begin{split}
\mathcal{F}\left[\delta(\vec{k}-\vec{k}_0) \right]&=e^{ i \vec{k}_0 \vec{x}}
\\
&=\psi(\vec{x})
\end{split}
\end{equation}

Such that:
\begin{equation}
\begin{split}
-i \hbar \frac{ \partial}{\partial \vec{x}} \psi(\vec{x})&=-i \hbar \frac{ \partial}{\partial \vec{x}} e^{ i \vec{k}_0 \vec{x}}
\\
&= \vec{k}_0 e^{ i \vec{k}_0 \vec{x}}
\\
&= \vec{k}_0 \psi(\vec{x})
\end{split}
\end{equation}

 

[PeterDonis]

My understanding of a plane wave is that it has non-zero amplitudes for many values of $\vec{k}$.

I don't know where you're getting that from. A plane wave is a momentum eigenstate. That means it has only one value for momentum, hence only one value for $\vec{k}$.

I would assume the following is true:

ϕ(⃗k)​

$$\begin{equation} \begin{split} \phi(\vec{k})&= \delta(\vec{k}-\vec{k}_0) \end{split} \end{equation}$$​

Yes.

Such that

Not quite. The eigenvalue equation, as I said in my previous post, is

$$
\hat{p} \psi(\vec{x}) = - i \hbar \frac{\partial}{\partial \vec{x}} \psi(\vec{x}) = \hbar \vec{k} \psi(\vec{x})
$$

This is from the definition of momentum as $\hbar k$, the definition of an eigenstate/eigenvalue, and the Fourier transform. The function you wrote down does not solve that equation. But it is close.

 

[redtree]

As you may be noticing, I am trying to avoid utilizing the Hamiltonian in the derivation. It seems to me that the Hamiltonian assumes the very momentum equivalence (between $\hbar \vec{k}$ and $m \vec{v}$) that I am trying to prove.

As to the following:
\begin{equation}
\begin{split}
\hat{E}&= i \hbar \frac{\partial}{\partial t}
\end{split}
\end{equation}

For the purposes of our discussion, I derive that from the wave equation for a free particle as follows:
\begin{equation}
\begin{split}
\psi(\vec{x},t)&=A e^{i (\vec{k}\vec{x}-\omega t)}
\end{split}
\end{equation}

Such that:
\begin{equation}
\begin{split}
i \hbar \frac{\partial}{\partial t} \left[\psi(\vec{x},t)\right]&= i \hbar \frac{\partial}{\partial t}\left[A e^{i (\vec{k}\vec{x}-\omega t)} \right]
\\
&=\hbar \omega \psi(\vec{x},t)
\\
&=E \psi(\vec{x},t)
\end{split}
\end{equation}

 

[PeterDonis]

It seems to me that the Hamiltonian assumes the very momentum equivalence (between $\hbar \vec{k}$ and $m \vec{v}$) that I am trying to prove.

I don't know why you would think that. The Hamiltonian for a free particle doesn't mention anything at all about waves.

For the purposes of our discussion, I derive that from the wave equation for a free particle

What wave equation for a free particle? A free particle isn't a wave. At least, you can write down its Hamiltonian without having to say anything about waves, as I said above. And if you're looking for a way to derive $\hbar \vec{k} = m \vec{v}$, how do you expect to do it if you refuse to look at any formula that doesn't have $\vec{k}$ in it?

 

[redtree]

If you're going to use the Hamiltonian as a measure of $\hbar \omega$, don't you have to prove that? It's the same problem as assuming $\hbar \vec{k}= m \vec{v}$, i.e., $\hbar \omega = \frac{m \vec{v}^2}{2} + \vec{V}$.

Any formula that doesn't have $\vec{k}$ (or $\omega$ for that matter) needs to be related to $\vec{k}$ (or $\omega$), presumably via the Fourier transform.

 

[PeterDonis]

If you're going to use the Hamiltonian as a measure of $\hbar \omega$,

Who said we are going to do that? You keep on reading things into my posts that I haven't said. Write down the Hamiltonian for a free particle in terms of the momentum $p$, and then take its derivative with respect to $p$. What do you get?

 

[PeterDonis]

Also, with regard to the formula $E = i \hbar \partial / \partial t$ that you wrote down. When you write down the Hamiltonian for a free particle, it is an operator, and since you have it in terms of $p$, you can write down what the operator is in the position representation. What do you get? Hint: it's not $i \hbar \partial / \partial t$.

 

[redtree]

The Hamiltonian:
\begin{equation}
\begin{split}
\mathcal{H}&= \frac{\vec{p}^2}{2 m}+ \vec{V}
\end{split}
\end{equation}

Such that:
\begin{equation}
\begin{split}
\frac{\partial}{\partial \vec{p}}\left[\mathcal{H}\right]&= \frac{\vec{p}}{m}+ \frac{\partial \vec{V}}{\partial \vec{p}}
\end{split}
\end{equation}

 

[redtree]

You are right. Sorry. I was rushing.

 

[redtree]

For a free particle:
$\frac{\partial \mathcal{H}}{\partial \vec{p}}=\frac{\vec{p}}{m}$

 

[PeterDonis]

You are right.

Ok, I see you edited the post. So, we have that $dE / d\vec{p} = \vec{p} / m$. And this describes a free particle; we haven't said anything about waves. But if we treat $dE / d\vec{p}$ as the group velocity of a wave associated with the particle--such as a quantum wave function describing the particle--then we have that $\vec{v} = \vec{p} / m$, or $\vec{p} = m \vec{v}$. Now just equate that with the other formula for $\vec{p}$ that you already know from the rest of the stuff we did above.

 

[redtree]

Yes; that's good, but in this derivation, group velocity has been defined in terms of $\omega$ and $\vec{k}$ not $\vec{x}$ and $t$, where $\vec{v}_P=\frac{\omega}{\vec{k}}$ and $\vec{v}_G=\frac{\partial \omega}{\partial \vec{k}}$.

 

[redtree]

This derivation also utilizes two separate definitions of total energy $E = \hbar \omega$ and $E =\mathcal{H}$ and equates them without deriving the equality, i.e. $\hbar \omega = \mathcal{H}$.

 

[PeterDonis]

in this derivation, group velocity has been defined in terms of $\omega$ and $\vec{k}$

No, it hasn't, it's been defined in terms of $E$ and $p$. Those have representations in both momentum space and position space. You are focusing on the momentum space representations, but actually what I said is independent of any representation. Properly viewed, it's a relationship between operators; it's telling you that the wave number operator $\hat{k}$ must have a certain relationship to the group velocity operator $\hat{v}$. (A more rigorous derivation would also show that the group velocity operator, in the appropriate classical limit, gives the classical velocity of the free particle. A really rigorous derivation would also not use plane waves but wave packets, and would express the relationship in terms of expectation values.)

This derivation also utilizes two separate definitions of total energy $E = \hbar \omega$ and $E =\mathcal{H}$

No, it doesn't. It just defines "energy" as the Hamiltonian operator. It doesn't say anything about any particular representation of it. See above.

 

[redtree]

I get your point about operators having representations in position and momentum space, such that in whatever space one is working, the operator produces the observable. Thus, the wavenumber operator in position space is $-i \frac{\partial}{\partial \vec{x}}$ while in momentum space it is simply $\vec{k}$. The momentum operator is simply $\hbar$ times the wavenumber operator $\hat{\vec{k}}$. Thus, in either momentum or position space, the momentum operator is a function of the wavenumber operator. The position operator is essentially the opposite of the wavenumber operator, such that in position space $\hat{\vec{x}}=\vec{x}$ and in momentum space $\hat{\vec{x}}=i \frac{\partial}{\partial \vec{k}}$. I still don't see how momentum can be a function of group velocity unless group velocity is defined in terms of $\vec{k}$ and $\omega$. Where is there a form of the momentum operator where $\hat{\vec{p}}$ is a function of the position operator $\hat{\vec{x}}$?

 

[PeterDonis]

Where is there a form of the momentum operator where $\hat{\vec{p}}$ is a function of the position operator $\hat{\vec{x}}$?

There isn't. Neither of those operators is a function of the other.

I still don't see how momentum can be a function of group velocity

That's not what we showed. We showed that group velocity is a function of momentum--specifically, it's $\hat{p} / m$. Again, this is independent of the representation we choose for the operators.